Subgroup

Definition: Subgroup

Let $G$ be a Group. Then a Subgroup $H$ of $G$ is a subset that

i. Contains the identity, ii. Closed under Group Multiplication, iii. and Closed under taking inverses

$H$ is then a Group (inheriting the group law from $G$).

Finding Subgroups

Instead of looking at every subset of a group, you can use the idea of generators. In essence, substitute in one element of the group and see what set ends up forming under the binary operation.

Lagrange’s Theorem